Question

Points \(P(6, 0)\), \(Q(2, 8)\) and \(R(-2, 4)\) are vertices of \(\Delta PQR\). It is given that \(MN \parallel QR\) such that \(\frac{PM}{MQ} = \frac{1}{3}\). Using distance formula and ratio formula, show that \(\frac{MN}{QR} = \frac{1}{4}\).

Hint:

Alternatively, you can prove this using similar triangles.
Since \(MN \parallel QR\), we have \(\Delta PMN \sim \Delta PQR\) by AA similarity.
Therefore, the ratio of any two corresponding sides is equal to the ratio of any other two corresponding sides:
\[ \frac{MN}{QR} = \frac{PM}{PQ} \]
Since \(\frac{PM}{MQ} = \frac{1}{3}\), we have \(\frac{PM}{PQ} = \frac{1}{1 + 3} = \frac{1}{4}\).
Thus, \(\frac{MN}{QR} = \frac{1}{4}\) directly without finding coordinates or using the distance formula!

Answer 1

Correct Answer: 4

Step 1: Understand the Setup.
Triangle $PQR$ has vertices $P(6,0)$, $Q(2,8)$, $R(-2,4)$. $MN \parallel QR$ with $M$ on $PQ$ and $N$ on $PR$. Given $\dfrac{PM}{MQ} = \dfrac{1}{3}$, we need to show $\dfrac{MN}{QR} = \dfrac{1}{4}$.
Step 2: Find Coordinates of M Using the Section Formula.
$M$ divides $PQ$ in ratio $PM : MQ = 1 : 3$ (from $P$). By the section formula: \[ M = \left(\frac{1 \times x_Q + 3 \times x_P}{1+3},\ \frac{1 \times y_Q + 3 \times y_P}{1+3}\right) = \left(\frac{2 + 18}{4},\ \frac{8 + 0}{4}\right) = (5,\ 2) \]
Step 3: Find Coordinates of N Using the Section Formula.
Since $MN \parallel QR$ and $\dfrac{PM}{PQ} = \dfrac{1}{4}$, point $N$ divides $PR$ in the same ratio $1:3$ from $P$: \[ N = \left(\frac{1 \times (-2) + 3 \times 6}{4},\ \frac{1 \times 4 + 3 \times 0}{4}\right) = \left(\frac{-2+18}{4},\ \frac{4}{4}\right) = (4,\ 1) \]
Step 4: Calculate Length MN Using Distance Formula.
\[ MN = \sqrt{(5-4)^2 + (2-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Step 5: Calculate Length QR Using Distance Formula.
\[ QR = \sqrt{(2-(-2))^2 + (8-4)^2} = \sqrt{4^2 + 4^2} = \sqrt{16+16} = \sqrt{32} = 4\sqrt{2} \]
Step 6: Compute the Ratio and Conclude.
\[ \frac{MN}{QR} = \frac{\sqrt{2}}{4\sqrt{2}} = \frac{1}{4} \] Hence proved. \[ \boxed{\dfrac{MN}{QR} = \dfrac{1}{4}} \]